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Flat knot 6.367

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,2,2,3,4,1,1,2,2,2,2,2,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.367']
Arrow polynomial of the knot is: -2K1K2 - 4K1K3 + K1 + 2K2 + K3 + 2K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147']
Outer characteristic polynomial of the knot is: t^7+63t^5+46t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.367']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 480K1*4K2 - 2704K14 + 1760K1*3K2K3 - 704K1*3K3 + 160K12K2*3 - 3232K1*2K2*2 + 128K1*2K2K32 - 1344K1*2K2K4 + 6160K1*2K2 - 2096K12K3*2 - 288K1*2K3K5 - 64K1*2K4*2 - 64K1*2K5*2 - 4164K1*2 + 64K1K23K3 + 128K1K2*2K3K4 - 1152K1K22K3 + 64K1K2*2K4K5 - 128K1K22K5 - 128K1K2K3K4 - 96K1K2K3K6 + 6376K1K2K3 - 64K1K2K4K5 - 64K1K2K5K6 + 3024K1K3K4 + 536K1K4K5 + 168K1K5K6 + 24K1K6K7 - 96K2*4 - 32K2*3K6 - 304K22K3*2 - 168K2*2K4*2 + 1160K2*2K4 - 80K22K5*2 - 16K2*2K6*2 - 3934K2*2 + 712K2K3K5 + 256K2K4K6 + 120K2K5K7 + 32K2K6K8 + 72K3*2K6 - 2532K32 - 1204K4*2 - 468K5*2 - 170K6*2 - 44K7*2 - 12K8**2 + 4182
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.367']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4661', 'vk6.4950', 'vk6.6115', 'vk6.6604', 'vk6.8124', 'vk6.8528', 'vk6.9510', 'vk6.9867', 'vk6.20375', 'vk6.21716', 'vk6.27687', 'vk6.29231', 'vk6.39123', 'vk6.41377', 'vk6.45867', 'vk6.47528', 'vk6.48693', 'vk6.48898', 'vk6.49449', 'vk6.49670', 'vk6.50705', 'vk6.50906', 'vk6.51192', 'vk6.51395', 'vk6.57244', 'vk6.58469', 'vk6.61870', 'vk6.63005', 'vk6.66867', 'vk6.67735', 'vk6.69491', 'vk6.70213']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,2,2,3,4,1,1,2,2,2,2,2,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.367']

Arrow polynomial of the knot is: -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147']

Outer characteristic polynomial of the knot is: t^7+63t^5+46t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.367']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 2704*K1**4 + 1760*K1**3*K2*K3 - 704*K1**3*K3 + 160*K1**2*K2**3 - 3232*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 1344*K1**2*K2*K4 + 6160*K1**2*K2 - 2096*K1**2*K3**2 - 288*K1**2*K3*K5 - 64*K1**2*K4**2 - 64*K1**2*K5**2 - 4164*K1**2 + 64*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1152*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6376*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K5*K6 + 3024*K1*K3*K4 + 536*K1*K4*K5 + 168*K1*K5*K6 + 24*K1*K6*K7 - 96*K2**4 - 32*K2**3*K6 - 304*K2**2*K3**2 - 168*K2**2*K4**2 + 1160*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 3934*K2**2 + 712*K2*K3*K5 + 256*K2*K4*K6 + 120*K2*K5*K7 + 32*K2*K6*K8 + 72*K3**2*K6 - 2532*K3**2 - 1204*K4**2 - 468*K5**2 - 170*K6**2 - 44*K7**2 - 12*K8**2 + 4182

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.367']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4661', 'vk6.4950', 'vk6.6115', 'vk6.6604', 'vk6.8124', 'vk6.8528', 'vk6.9510', 'vk6.9867', 'vk6.20375', 'vk6.21716', 'vk6.27687', 'vk6.29231', 'vk6.39123', 'vk6.41377', 'vk6.45867', 'vk6.47528', 'vk6.48693', 'vk6.48898', 'vk6.49449', 'vk6.49670', 'vk6.50705', 'vk6.50906', 'vk6.51192', 'vk6.51395', 'vk6.57244', 'vk6.58469', 'vk6.61870', 'vk6.63005', 'vk6.66867', 'vk6.67735', 'vk6.69491', 'vk6.70213']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is c.

The reverse -K is

The mirror image K* is

The reversed mirror image -K* is

The fillings (up to the first 10) associated to the algebraic genus:

Or click here to check the fillings

invariant	value
Gauss code	O1O2O3O4O5U3U5O6U2U1U4U6
R3 orbit	{'O1O2O3O4O5U3U5O6U2U1U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U2U5U4O6U1U3
Gauss code of K*	O1O2O3O4U2U1U5U3U6O5O6U4
Gauss code of -K*	O1O2O3O4U1O5O6U5U2U6U4U3
Diagrammatic symmetry type	c
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -2 -2 -2 2 1 3],[ 2 0 0 -1 3 1 3],[ 2 0 0 -1 2 1 2],[ 2 1 1 0 2 1 1],[-2 -3 -2 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix	[[ 0 3 2 1 -2 -2 -2],[-3 0 -1 0 -1 -2 -3],[-2 1 0 0 -2 -2 -3],[-1 0 0 0 -1 -1 -1],[ 2 1 2 1 0 1 1],[ 2 2 2 1 -1 0 0],[ 2 3 3 1 -1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,2,2,2,1,0,1,2,3,0,2,2,3,1,1,1,-1,-1,0]
Phi over symmetry	[-3,-2,-1,2,2,2,0,2,2,3,4,1,1,2,2,2,2,2,0,-1,-1]
Phi of -K	[-2,-2,-2,1,2,3,-1,-1,2,2,4,0,2,1,2,2,2,3,1,2,0]
Phi of K*	[-3,-2,-1,2,2,2,0,2,2,3,4,1,1,2,2,2,2,2,0,-1,-1]
Phi of -K*	[-2,-2,-2,1,2,3,-1,0,1,2,2,1,1,2,1,1,3,3,0,0,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+2t^2-t
Normalized Jones-Krushkal polynomial	5z^2+25z+31
Enhanced Jones-Krushkal polynomial	5w^3z^2+25w^2z+31w
Inner characteristic polynomial	t^6+37t^4+18t^2+1
Outer characteristic polynomial	t^7+63t^5+46t^3+7t
Flat arrow polynomial	-2K1K2 - 4K1K3 + K1 + 2K2 + K3 + 2K4 + 1
2-strand cable arrow polynomial	-256K14K2*2 + 480K1*4K2 - 2704K14 + 1760K1*3K2K3 - 704K1*3K3 + 160K12K2*3 - 3232K1*2K2*2 + 128K1*2K2K32 - 1344K1*2K2K4 + 6160K1*2K2 - 2096K12K3*2 - 288K1*2K3K5 - 64K1*2K4*2 - 64K1*2K5*2 - 4164K1*2 + 64K1K23K3 + 128K1K2*2K3K4 - 1152K1K22K3 + 64K1K2*2K4K5 - 128K1K22K5 - 128K1K2K3K4 - 96K1K2K3K6 + 6376K1K2K3 - 64K1K2K4K5 - 64K1K2K5K6 + 3024K1K3K4 + 536K1K4K5 + 168K1K5K6 + 24K1K6K7 - 96K2*4 - 32K2*3K6 - 304K22K3*2 - 168K2*2K4*2 + 1160K2*2K4 - 80K22K5*2 - 16K2*2K6*2 - 3934K2*2 + 712K2K3K5 + 256K2K4K6 + 120K2K5K7 + 32K2K6K8 + 72K3*2K6 - 2532K32 - 1204K4*2 - 468K5*2 - 170K6*2 - 44K7*2 - 12K8**2 + 4182
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {3, 4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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